Problem: If $\mathbf{A}^{-1} = \begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix},$ then find the inverse of $\mathbf{A}^2.$
Explanation: Note that $(\mathbf{A}^{-1})^2 \mathbf{A}^2 = \mathbf{A}^{-1} \mathbf{A}^{-1} \mathbf{A} \mathbf{A} = \mathbf{I},$ so the inverse of $\mathbf{A}^2$ is
\[(\mathbf{A}^{-1})^2 = \begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix}^2 = \boxed{\begin{pmatrix} -1 & -5 \\ 1 & 4 \end{pmatrix}}.\]